Related papers: Beyond Quantifier-Free Interpolation in Extensions…
Traditionally, research on Craig interpolation is concerned with (a) establishing the Craig interpolation property (CIP) of a logic saying that every valid implication in the logic has a Craig interpolant and (b) designing algorithms that…
We introduce Craig interpolation and related notions such as uniform interpolation, Beth definability, and theory decomposition in classical propositional logic. We present four approaches to computing interpolants: via quantifier…
Existing techniques for Craig interpolation for the quantifier-free fragment of the theory of arrays are inefficient for computing sequence and tree interpolants: the solver needs to run for every partitioning $(A, B)$ of the interpolation…
A modular proof-theoretic framework was recently developed to prove Craig interpolation for normal modal logics based on generalizations of sequent calculi (e.g., nested sequents, hypersequents, and labelled sequents). In this paper, we…
The first-order theory of addition over the natural numbers, known as Presburger arithmetic, is decidable in double exponential time. Adding an uninterpreted unary predicate to the language leads to an undecidable theory. We sharpen the…
Recent research has established complexity results for the problem of deciding the existence of interpolants in logics lacking the Craig Interpolation Property (CIP). The proof techniques developed so far are non-constructive, and no…
Craig interpolation is a widespread method in verification, with important applications such as Predicate Abstraction, CounterExample Guided Abstraction Refinement and Lazy Abstraction With Interpolants. Most state-of-the-art model checking…
We develop foundations for computing Craig interpolants and similar intermediates of two given formulas with first-order theorem provers that construct clausal tableaux. Provers that can be understood in this way include efficient…
We propose a notion of quantum control in a quantum programming language which permits the superposition of finitely many quantum operations without performing a measurement. This notion takes the form of a conditional construct similar to…
All known quantifier elimination procedures for Presburger arithmetic require doubly exponential time for eliminating a single block of existentially quantified variables. It has even been claimed in the literature that this upper bound is…
We identify a fragment of Presburger arithmetic enriched with free function symbols and cardinality constraints for interpreted sets, which is amenable to automated analysis. We establish decidability and complexity results for such a…
We have recently presented a general method of proving the fundamental logical properties of Craig and Lyndon Interpolation (IPs) by induction on derivations in a wide class of internal sequent calculi, including sequents, hypersequents,…
Nonlinear interpolants have been shown useful for the verification of programs and hybrid systems in contexts of theorem proving, model checking, abstract interpretation, etc. The underlying synthesis problem, however, is challenging and…
Interpolation based software model checkers have been successfully employed to automatically prove programs correct. Their power comes from interpolating SMT solvers that check the feasibility of potential counterexamples and compute…
We give a quantifier elimination procedures for the extension of Presburger arithmetic with a unary threshold counting quantifier $\exists^{\ge c} y$ that determines whether the number of different $y$ satisfying some formula is at least $c…
Craig's interpolation theorem (Craig 1957) is an important theorem known for propositional logic and first-order logic. It says that if a logical formula $\beta$ logically follows from a formula $\alpha$, then there is a formula $\gamma$,…
Interpolation is an essential tool in software verification, where first-order theories are used to constrain datatypes manipulated by programs. In this paper, we introduce the datatype theory of contiguous arrays with maxdiff, where arrays…
We consider Presburger arithmetic (PA) extended by scalar multiplication by an algebraic irrational number $\alpha$, and call this extension $\alpha$-Presburger arithmetic ($\alpha$-PA). We show that the complexity of deciding sentences in…
We develop foundations for computing Craig-Lyndon interpolants of two given formulas with first-order theorem provers that construct clausal tableaux. Provers that can be understood in this way include efficient machine-oriented systems…
In the present paper, we consider Presburger arithmetic PrA and the theory of real closed fields RCF. Due to quantifier elimination in these theories, there are two kinds of natural ways to axiomatize them. Namely, on one hand, PrA can be…